If $\theta=\frac{-5 \pi}{4}$, then
\[
\sin (\theta)=
\]
\[
\cos (\theta)=
\]
Give exact values. No decimals allowed!
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Answer
The exact value of \(\sin(\theta)\) is \(\boxed{\frac{\sqrt{2}}{2}}\) and the exact value of \(\cos(\theta)\) is \(\boxed{-\frac{\sqrt{2}}{2}}\).
Step 1 ：The angle \(\frac{-5 \pi}{4}\) is equivalent to \(\frac{3 \pi}{4}\) in the unit circle because they are coterminal angles (angles that share the same terminal side).
Step 2 ：The point on the unit circle that corresponds to the angle \(\frac{3 \pi}{4}\) is \(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\).
Step 3 ：Therefore, the cosine of the angle is \(-\frac{\sqrt{2}}{2}\) and the sine of the angle is \(\frac{\sqrt{2}}{2}\).
Step 4 ：The exact value of \(\sin(\theta)\) is \(\boxed{\frac{\sqrt{2}}{2}}\) and the exact value of \(\cos(\theta)\) is \(\boxed{-\frac{\sqrt{2}}{2}}\).
